LMFDB - Embedded newform 5175.2.a.bv.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5175,2,Mod(1,5175)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5175.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$41.3225830460$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.37988$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37988 q^{2} +3.66382 q^{4} -2.28394 q^{7} +3.95969 q^{8} +O(q^{10})$	$q+2.37988 q^{2} +3.66382 q^{4} -2.28394 q^{7} +3.95969 q^{8} -1.12432 q^{11} -5.95969 q^{13} -5.43550 q^{14} +2.09594 q^{16} +5.80007 q^{17} -4.08401 q^{19} -2.67575 q^{22} -1.00000 q^{23} -14.1833 q^{26} -8.36795 q^{28} +0.408263 q^{29} -3.19187 q^{31} -2.93130 q^{32} +13.8035 q^{34} +9.80345 q^{37} -9.71944 q^{38} -6.27087 q^{41} -7.75474 q^{43} -4.11931 q^{44} -2.37988 q^{46} -6.40020 q^{47} -1.78361 q^{49} -21.8352 q^{52} -6.73590 q^{53} -9.04370 q^{56} +0.971615 q^{58} +4.75976 q^{59} -6.33265 q^{61} -7.59627 q^{62} -11.1680 q^{64} -0.283942 q^{67} +21.2504 q^{68} +13.9516 q^{71} -9.61659 q^{73} +23.3310 q^{74} -14.9631 q^{76} +2.56788 q^{77} +4.48387 q^{79} -14.9239 q^{82} -10.8223 q^{83} -18.4553 q^{86} -4.45196 q^{88} -5.68414 q^{89} +13.6116 q^{91} -3.66382 q^{92} -15.2317 q^{94} -11.0676 q^{97} -4.24477 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8	$ 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8} - 2 q^{11} - 14 q^{13} + 4 q^{14} + 2 q^{16} + 14 q^{17} - 4 q^{19} - 4 q^{22} - 4 q^{23} - 6 q^{26} - 18 q^{28} - 4 q^{29} + 2 q^{32} + 14 q^{34} - 2 q^{37} - 10 q^{38} + 8 q^{41} - 4 q^{43} - 6 q^{44} + 2 q^{47} - 18 q^{52} + 4 q^{53} - 14 q^{56} - 8 q^{61} - 28 q^{62} - 20 q^{64} + 2 q^{67} + 8 q^{68} + 24 q^{71} - 18 q^{73} + 36 q^{74} - 18 q^{76} + 4 q^{77} + 24 q^{79} - 32 q^{82} - 6 q^{83} - 14 q^{86} + 10 q^{88} + 8 q^{89} + 26 q^{91} - 2 q^{92} - 42 q^{94} - 34 q^{97} - 28 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8 - 2 * q^11 - 14 * q^13 + 4 * q^14 + 2 * q^16 + 14 * q^17 - 4 * q^19 - 4 * q^22 - 4 * q^23 - 6 * q^26 - 18 * q^28 - 4 * q^29 + 2 * q^32 + 14 * q^34 - 2 * q^37 - 10 * q^38 + 8 * q^41 - 4 * q^43 - 6 * q^44 + 2 * q^47 - 18 * q^52 + 4 * q^53 - 14 * q^56 - 8 * q^61 - 28 * q^62 - 20 * q^64 + 2 * q^67 + 8 * q^68 + 24 * q^71 - 18 * q^73 + 36 * q^74 - 18 * q^76 + 4 * q^77 + 24 * q^79 - 32 * q^82 - 6 * q^83 - 14 * q^86 + 10 * q^88 + 8 * q^89 + 26 * q^91 - 2 * q^92 - 42 * q^94 - 34 * q^97 - 28 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.37988	1.68283	0.841414	−	0.540391i	$-0.181724\pi$
$2$	2.37988	1.68283	0.841414	+	0.540391i	$0.181724\pi$
$3$	0	0
$3$	0	0
$4$	3.66382	1.83191
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.28394	−0.863249	−0.431624	−	0.902053i	$-0.642059\pi$
$7$	−2.28394	−0.863249	−0.431624	+	0.902053i	$0.642059\pi$
$8$	3.95969	1.39996
$9$	0	0
$10$	0	0
$11$	−1.12432	−0.338996	−0.169498	−	0.985531i	$-0.554215\pi$
$11$	−1.12432	−0.338996	−0.169498	+	0.985531i	$0.554215\pi$
$12$	0	0
$13$	−5.95969	−1.65292	−0.826460	−	0.562995i	$-0.809649\pi$
$13$	−5.95969	−1.65292	−0.826460	+	0.562995i	$0.809649\pi$
$14$	−5.43550	−1.45270
$15$	0	0
$16$	2.09594	0.523984
$17$	5.80007	1.40672	0.703362	−	0.710832i	$-0.251682\pi$
$17$	5.80007	1.40672	0.703362	+	0.710832i	$0.251682\pi$
$18$	0	0
$19$	−4.08401	−0.936936	−0.468468	−	0.883480i	$-0.655194\pi$
$19$	−4.08401	−0.936936	−0.468468	+	0.883480i	$0.655194\pi$
$20$	0	0
$21$	0	0
$22$	−2.67575	−0.570471
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	−14.1833	−2.78158
$27$	0	0
$28$	−8.36795	−1.58139
$29$	0.408263	0.0758125	0.0379062	−	0.999281i	$-0.487931\pi$
$29$	0.408263	0.0758125	0.0379062	+	0.999281i	$0.487931\pi$
$30$	0	0
$31$	−3.19187	−0.573277	−0.286639	−	0.958039i	$-0.592538\pi$
$31$	−3.19187	−0.573277	−0.286639	+	0.958039i	$0.592538\pi$
$32$	−2.93130	−0.518186
$33$	0	0
$34$	13.8035	2.36727
$35$	0	0
$36$	0	0
$37$	9.80345	1.61168	0.805839	−	0.592135i	$-0.201715\pi$
$37$	9.80345	1.61168	0.805839	+	0.592135i	$0.201715\pi$
$38$	−9.71944	−1.57670
$39$	0	0
$40$	0	0
$41$	−6.27087	−0.979345	−0.489673	−	0.871906i	$-0.662884\pi$
$41$	−6.27087	−0.979345	−0.489673	+	0.871906i	$0.662884\pi$
$42$	0	0
$43$	−7.75474	−1.18259	−0.591294	−	0.806456i	$-0.701383\pi$
$43$	−7.75474	−1.18259	−0.591294	+	0.806456i	$0.701383\pi$
$44$	−4.11931	−0.621009
$45$	0	0
$46$	−2.37988	−0.350894
$47$	−6.40020	−0.933566	−0.466783	−	0.884372i	$-0.654587\pi$
$47$	−6.40020	−0.933566	−0.466783	+	0.884372i	$0.654587\pi$
$48$	0	0
$49$	−1.78361	−0.254801
$50$	0	0
$51$	0	0
$52$	−21.8352	−3.02800
$53$	−6.73590	−0.925247	−0.462624	−	0.886555i	$-0.653092\pi$
$53$	−6.73590	−0.925247	−0.462624	+	0.886555i	$0.653092\pi$
$54$	0	0
$55$	0	0
$56$	−9.04370	−1.20851
$57$	0	0
$58$	0.971615	0.127579
$59$	4.75976	0.619667	0.309834	−	0.950791i	$-0.399727\pi$
$59$	4.75976	0.619667	0.309834	+	0.950791i	$0.399727\pi$
$60$	0	0
$61$	−6.33265	−0.810813	−0.405406	−	0.914137i	$-0.632870\pi$
$61$	−6.33265	−0.810813	−0.405406	+	0.914137i	$0.632870\pi$
$62$	−7.59627	−0.964727
$63$	0	0
$64$	−11.1680	−1.39600
$65$	0	0
$66$	0	0
$67$	−0.283942	−0.0346890	−0.0173445	−	0.999850i	$-0.505521\pi$
$67$	−0.283942	−0.0346890	−0.0173445	+	0.999850i	$0.505521\pi$
$68$	21.2504	2.57699
$69$	0	0
$70$	0	0
$71$	13.9516	1.65575	0.827877	−	0.560910i	$-0.189549\pi$
$71$	13.9516	1.65575	0.827877	+	0.560910i	$0.189549\pi$
$72$	0	0
$73$	−9.61659	−1.12554	−0.562769	−	0.826615i	$-0.690264\pi$
$73$	−9.61659	−1.12554	−0.562769	+	0.826615i	$0.690264\pi$
$74$	23.3310	2.71218
$75$	0	0
$76$	−14.9631	−1.71638
$77$	2.56788	0.292637
$78$	0	0
$79$	4.48387	0.504475	0.252238	−	0.967665i	$-0.418834\pi$
$79$	4.48387	0.504475	0.252238	+	0.967665i	$0.418834\pi$
$80$	0	0
$81$	0	0
$82$	−14.9239	−1.64807
$83$	−10.8223	−1.18790	−0.593951	−	0.804501i	$-0.702433\pi$
$83$	−10.8223	−1.18790	−0.593951	+	0.804501i	$0.702433\pi$
$84$	0	0
$85$	0	0
$86$	−18.4553	−1.99009
$87$	0	0
$88$	−4.45196	−0.474581
$89$	−5.68414	−0.602518	−0.301259	−	0.953542i	$-0.597407\pi$
$89$	−5.68414	−0.602518	−0.301259	+	0.953542i	$0.597407\pi$
$90$	0	0
$91$	13.6116	1.42688
$92$	−3.66382	−0.381980
$93$	0	0
$94$	−15.2317	−1.57103
$95$	0	0
$96$	0	0
$97$	−11.0676	−1.12374	−0.561870	−	0.827226i	$-0.689918\pi$
$97$	−11.0676	−1.12374	−0.561870	+	0.827226i	$0.689918\pi$
$98$	−4.24477	−0.428787
$99$	0	0
$100$	0	0
$101$	−1.60014	−0.159219	−0.0796097	−	0.996826i	$-0.525367\pi$
$101$	−1.60014	−0.159219	−0.0796097	+	0.996826i	$0.525367\pi$
$102$	0	0
$103$	−1.23218	−0.121411	−0.0607054	−	0.998156i	$-0.519335\pi$
$103$	−1.23218	−0.121411	−0.0607054	+	0.998156i	$0.519335\pi$
$104$	−23.5985	−2.31402
$105$	0	0
$106$	−16.0306	−1.55703
$107$	−0.235232	−0.0227408	−0.0113704	−	0.999935i	$-0.503619\pi$
$107$	−0.235232	−0.0227408	−0.0113704	+	0.999935i	$0.503619\pi$
$108$	0	0
$109$	7.43550	0.712192	0.356096	−	0.934449i	$-0.384108\pi$
$109$	7.43550	0.712192	0.356096	+	0.934449i	$0.384108\pi$
$110$	0	0
$111$	0	0
$112$	−4.78700	−0.452329
$113$	−2.28394	−0.214855	−0.107428	−	0.994213i	$-0.534261\pi$
$113$	−2.28394	−0.214855	−0.107428	+	0.994213i	$0.534261\pi$
$114$	0	0
$115$	0	0
$116$	1.49580	0.138882
$117$	0	0
$118$	11.3276	1.04279
$119$	−13.2470	−1.21435
$120$	0	0
$121$	−9.73590	−0.885082
$122$	−15.0709	−1.36446
$123$	0	0
$124$	−11.6944	−1.05019
$125$	0	0
$126$	0	0
$127$	10.2151	0.906444	0.453222	−	0.891398i	$-0.350275\pi$
$127$	10.2151	0.906444	0.453222	+	0.891398i	$0.350275\pi$
$128$	−20.7159	−1.83105
$129$	0	0
$130$	0	0
$131$	16.2232	1.41742	0.708712	−	0.705498i	$-0.249276\pi$
$131$	16.2232	1.41742	0.708712	+	0.705498i	$0.249276\pi$
$132$	0	0
$133$	9.32764	0.808809
$134$	−0.675747	−0.0583756
$135$	0	0
$136$	22.9665	1.96936
$137$	−4.89715	−0.418392	−0.209196	−	0.977874i	$-0.567085\pi$
$137$	−4.89715	−0.418392	−0.209196	+	0.977874i	$0.567085\pi$
$138$	0	0
$139$	4.43889	0.376502	0.188251	−	0.982121i	$-0.439718\pi$
$139$	4.43889	0.376502	0.188251	+	0.982121i	$0.439718\pi$
$140$	0	0
$141$	0	0
$142$	33.2032	2.78635
$143$	6.70060	0.560333
$144$	0	0
$145$	0	0
$146$	−22.8863	−1.89409
$147$	0	0
$148$	35.9181	2.95245
$149$	2.78361	0.228042	0.114021	−	0.993478i	$-0.463627\pi$
$149$	2.78361	0.228042	0.114021	+	0.993478i	$0.463627\pi$
$150$	0	0
$151$	11.9278	0.970669	0.485334	−	0.874329i	$-0.338698\pi$
$151$	11.9278	0.970669	0.485334	+	0.874329i	$0.338698\pi$
$152$	−16.1714	−1.31167
$153$	0	0
$154$	6.11125	0.492459
$155$	0	0
$156$	0	0
$157$	−22.9550	−1.83201	−0.916005	−	0.401167i	$-0.868605\pi$
$157$	−22.9550	−1.83201	−0.916005	+	0.401167i	$0.868605\pi$
$158$	10.6711	0.848945
$159$	0	0
$160$	0	0
$161$	2.28394	0.180000
$162$	0	0
$163$	10.2083	0.799578	0.399789	−	0.916607i	$-0.369083\pi$
$163$	10.2083	0.799578	0.399789	+	0.916607i	$0.369083\pi$
$164$	−22.9753	−1.79407
$165$	0	0
$166$	−25.7557	−1.99903
$167$	6.84715	0.529849	0.264924	−	0.964269i	$-0.414653\pi$
$167$	6.84715	0.529849	0.264924	+	0.964269i	$0.414653\pi$
$168$	0	0
$169$	22.5179	1.73215
$170$	0	0
$171$	0	0
$172$	−28.4120	−2.16639
$173$	4.29539	0.326572	0.163286	−	0.986579i	$-0.447791\pi$
$173$	4.29539	0.326572	0.163286	+	0.986579i	$0.447791\pi$
$174$	0	0
$175$	0	0
$176$	−2.35651	−0.177628
$177$	0	0
$178$	−13.5276	−1.01393
$179$	14.8626	1.11088	0.555442	−	0.831555i	$-0.312549\pi$
$179$	14.8626	1.11088	0.555442	+	0.831555i	$0.312549\pi$
$180$	0	0
$181$	13.1311	0.976027	0.488013	−	0.872836i	$-0.337722\pi$
$181$	13.1311	0.976027	0.488013	+	0.872836i	$0.337722\pi$
$182$	32.3939	2.40120
$183$	0	0
$184$	−3.95969	−0.291912
$185$	0	0
$186$	0	0
$187$	−6.52114	−0.476873
$188$	−23.4492	−1.71021
$189$	0	0
$190$	0	0
$191$	−13.9819	−1.01170	−0.505848	−	0.862623i	$-0.668820\pi$
$191$	−13.9819	−1.01170	−0.505848	+	0.862623i	$0.668820\pi$
$192$	0	0
$193$	−8.71267	−0.627152	−0.313576	−	0.949563i	$-0.601527\pi$
$193$	−8.71267	−0.627152	−0.313576	+	0.949563i	$0.601527\pi$
$194$	−26.3394	−1.89106
$195$	0	0
$196$	−6.53483	−0.466773
$197$	−22.4876	−1.60218	−0.801088	−	0.598547i	$-0.795745\pi$
$197$	−22.4876	−1.60218	−0.801088	+	0.598547i	$0.795745\pi$
$198$	0	0
$199$	9.01078	0.638757	0.319379	−	0.947627i	$-0.396526\pi$
$199$	9.01078	0.638757	0.319379	+	0.947627i	$0.396526\pi$
$200$	0	0
$201$	0	0
$202$	−3.80813	−0.267939
$203$	−0.932448	−0.0654450
$204$	0	0
$205$	0	0
$206$	−2.93245	−0.204313
$207$	0	0
$208$	−12.4911	−0.866104
$209$	4.59174	0.317617
$210$	0	0
$211$	5.60014	0.385529	0.192765	−	0.981245i	$-0.438255\pi$
$211$	5.60014	0.385529	0.192765	+	0.981245i	$0.438255\pi$
$212$	−24.6791	−1.69497
$213$	0	0
$214$	−0.559824	−0.0382688
$215$	0	0
$216$	0	0
$217$	7.29005	0.494881
$218$	17.6956	1.19850
$219$	0	0
$220$	0	0
$221$	−34.5666	−2.32520
$222$	0	0
$223$	7.06517	0.473119	0.236559	−	0.971617i	$-0.423980\pi$
$223$	7.06517	0.473119	0.236559	+	0.971617i	$0.423980\pi$
$224$	6.69493	0.447324
$225$	0	0
$226$	−5.43550	−0.361564
$227$	25.1072	1.66643	0.833213	−	0.552952i	$-0.186499\pi$
$227$	25.1072	1.66643	0.833213	+	0.552952i	$0.186499\pi$
$228$	0	0
$229$	−3.20366	−0.211704	−0.105852	−	0.994382i	$-0.533757\pi$
$229$	−3.20366	−0.211704	−0.105852	+	0.994382i	$0.533757\pi$
$230$	0	0
$231$	0	0
$232$	1.61659	0.106135
$233$	18.6388	1.22107	0.610535	−	0.791989i	$-0.290954\pi$
$233$	18.6388	1.22107	0.610535	+	0.791989i	$0.290954\pi$
$234$	0	0
$235$	0	0
$236$	17.4389	1.13518
$237$	0	0
$238$	−31.5263	−2.04355
$239$	−3.18185	−0.205817	−0.102908	−	0.994691i	$-0.532815\pi$
$239$	−3.18185	−0.205817	−0.102908	+	0.994691i	$0.532815\pi$
$240$	0	0
$241$	4.79844	0.309095	0.154547	−	0.987985i	$-0.450608\pi$
$241$	4.79844	0.309095	0.154547	+	0.987985i	$0.450608\pi$
$242$	−23.1703	−1.48944
$243$	0	0
$244$	−23.2017	−1.48534
$245$	0	0
$246$	0	0
$247$	24.3394	1.54868
$248$	−12.6388	−0.802566
$249$	0	0
$250$	0	0
$251$	2.24668	0.141809	0.0709045	−	0.997483i	$-0.477411\pi$
$251$	2.24668	0.141809	0.0709045	+	0.997483i	$0.477411\pi$
$252$	0	0
$253$	1.12432	0.0706855
$254$	24.3107	1.52539
$255$	0	0
$256$	−26.9653	−1.68533
$257$	11.0148	0.687086	0.343543	−	0.939137i	$-0.388373\pi$
$257$	11.0148	0.687086	0.343543	+	0.939137i	$0.388373\pi$
$258$	0	0
$259$	−22.3905	−1.39128
$260$	0	0
$261$	0	0
$262$	38.6092	2.38528
$263$	−11.0585	−0.681898	−0.340949	−	0.940082i	$-0.610748\pi$
$263$	−11.0585	−0.681898	−0.340949	+	0.940082i	$0.610748\pi$
$264$	0	0
$265$	0	0
$266$	22.1986	1.36109
$267$	0	0
$268$	−1.04031	−0.0635471
$269$	9.90392	0.603853	0.301926	−	0.953331i	$-0.402370\pi$
$269$	9.90392	0.603853	0.301926	+	0.953331i	$0.402370\pi$
$270$	0	0
$271$	4.82655	0.293192	0.146596	−	0.989196i	$-0.453168\pi$
$271$	4.82655	0.293192	0.146596	+	0.989196i	$0.453168\pi$
$272$	12.1566	0.737100
$273$	0	0
$274$	−11.6546	−0.704081
$275$	0	0
$276$	0	0
$277$	−7.54303	−0.453217	−0.226608	−	0.973986i	$-0.572764\pi$
$277$	−7.54303	−0.453217	−0.226608	+	0.973986i	$0.572764\pi$
$278$	10.5640	0.633588
$279$	0	0
$280$	0	0
$281$	−6.01145	−0.358613	−0.179306	−	0.983793i	$-0.557385\pi$
$281$	−6.01145	−0.358613	−0.179306	+	0.983793i	$0.557385\pi$
$282$	0	0
$283$	15.6388	0.929631	0.464816	−	0.885407i	$-0.346121\pi$
$283$	15.6388	0.929631	0.464816	+	0.885407i	$0.346121\pi$
$284$	51.1163	3.03319
$285$	0	0
$286$	15.9466	0.942943
$287$	14.3223	0.845419
$288$	0	0
$289$	16.6408	0.978870
$290$	0	0
$291$	0	0
$292$	−35.2335	−2.06188
$293$	−3.21639	−0.187904	−0.0939518	−	0.995577i	$-0.529950\pi$
$293$	−3.21639	−0.187904	−0.0939518	+	0.995577i	$0.529950\pi$
$294$	0	0
$295$	0	0
$296$	38.8186	2.25629
$297$	0	0
$298$	6.62465	0.383756
$299$	5.95969	0.344658
$300$	0	0
$301$	17.7114	1.02087
$302$	28.3867	1.63347
$303$	0	0
$304$	−8.55982	−0.490940
$305$	0	0
$306$	0	0
$307$	34.4702	1.96732	0.983659	−	0.180044i	$-0.0576240\pi$
$307$	34.4702	1.96732	0.983659	+	0.180044i	$0.0576240\pi$
$308$	9.40826	0.536086
$309$	0	0
$310$	0	0
$311$	18.7443	1.06289	0.531446	−	0.847092i	$-0.321649\pi$
$311$	18.7443	1.06289	0.531446	+	0.847092i	$0.321649\pi$
$312$	0	0
$313$	−9.91260	−0.560294	−0.280147	−	0.959957i	$-0.590383\pi$
$313$	−9.91260	−0.560294	−0.280147	+	0.959957i	$0.590383\pi$
$314$	−54.6301	−3.08296
$315$	0	0
$316$	16.4281	0.924153
$317$	−9.24024	−0.518984	−0.259492	−	0.965745i	$-0.583555\pi$
$317$	−9.24024	−0.518984	−0.259492	+	0.965745i	$0.583555\pi$
$318$	0	0
$319$	−0.459018	−0.0257001
$320$	0	0
$321$	0	0
$322$	5.43550	0.302909
$323$	−23.6875	−1.31801
$324$	0	0
$325$	0	0
$326$	24.2946	1.34555
$327$	0	0
$328$	−24.8307	−1.37105
$329$	14.6177	0.805899
$330$	0	0
$331$	−23.9684	−1.31742	−0.658712	−	0.752395i	$-0.728898\pi$
$331$	−23.9684	−1.31742	−0.658712	+	0.752395i	$0.728898\pi$
$332$	−39.6509	−2.17613
$333$	0	0
$334$	16.2954	0.891644
$335$	0	0
$336$	0	0
$337$	−9.76477	−0.531921	−0.265960	−	0.963984i	$-0.585689\pi$
$337$	−9.76477	−0.531921	−0.265960	+	0.963984i	$0.585689\pi$
$338$	53.5898	2.91490
$339$	0	0
$340$	0	0
$341$	3.58869	0.194338
$342$	0	0
$343$	20.0613	1.08321
$344$	−30.7064	−1.65558
$345$	0	0
$346$	10.2225	0.549565
$347$	1.38441	0.0743190	0.0371595	−	0.999309i	$-0.488169\pi$
$347$	1.38441	0.0743190	0.0371595	+	0.999309i	$0.488169\pi$
$348$	0	0
$349$	−32.3106	−1.72954	−0.864772	−	0.502164i	$-0.832537\pi$
$349$	−32.3106	−1.72954	−0.864772	+	0.502164i	$0.832537\pi$
$350$	0	0
$351$	0	0
$352$	3.29573	0.175663
$353$	11.4386	0.608813	0.304406	−	0.952542i	$-0.401542\pi$
$353$	11.4386	0.608813	0.304406	+	0.952542i	$0.401542\pi$
$354$	0	0
$355$	0	0
$356$	−20.8257	−1.10376
$357$	0	0
$358$	35.3712	1.86943
$359$	−9.95568	−0.525441	−0.262720	−	0.964872i	$-0.584620\pi$
$359$	−9.95568	−0.525441	−0.262720	+	0.964872i	$0.584620\pi$
$360$	0	0
$361$	−2.32087	−0.122151
$362$	31.2504	1.64248
$363$	0	0
$364$	49.8704	2.61392
$365$	0	0
$366$	0	0
$367$	−27.6785	−1.44481	−0.722403	−	0.691472i	$-0.756963\pi$
$367$	−27.6785	−1.44481	−0.722403	+	0.691472i	$0.756963\pi$
$368$	−2.09594	−0.109258
$369$	0	0
$370$	0	0
$371$	15.3844	0.798719
$372$	0	0
$373$	−28.7356	−1.48787	−0.743936	−	0.668251i	$-0.767043\pi$
$373$	−28.7356	−1.48787	−0.743936	+	0.668251i	$0.767043\pi$
$374$	−15.5195	−0.802495
$375$	0	0
$376$	−25.3428	−1.30696
$377$	−2.43312	−0.125312
$378$	0	0
$379$	−18.2715	−0.938546	−0.469273	−	0.883053i	$-0.655484\pi$
$379$	−18.2715	−0.938546	−0.469273	+	0.883053i	$0.655484\pi$
$380$	0	0
$381$	0	0
$382$	−33.2753	−1.70251
$383$	−29.0356	−1.48365	−0.741826	−	0.670593i	$-0.766040\pi$
$383$	−29.0356	−1.48365	−0.741826	+	0.670593i	$0.766040\pi$
$384$	0	0
$385$	0	0
$386$	−20.7351	−1.05539
$387$	0	0
$388$	−40.5495	−2.05859
$389$	−2.86423	−0.145222	−0.0726112	−	0.997360i	$-0.523133\pi$
$389$	−2.86423	−0.145222	−0.0726112	+	0.997360i	$0.523133\pi$
$390$	0	0
$391$	−5.80007	−0.293322
$392$	−7.06254	−0.356712
$393$	0	0
$394$	−53.5177	−2.69619
$395$	0	0
$396$	0	0
$397$	−16.5592	−0.831080	−0.415540	−	0.909575i	$-0.636407\pi$
$397$	−16.5592	−0.831080	−0.415540	+	0.909575i	$0.636407\pi$
$398$	21.4446	1.07492
$399$	0	0
$400$	0	0
$401$	3.09479	0.154547	0.0772733	−	0.997010i	$-0.475379\pi$
$401$	3.09479	0.154547	0.0772733	+	0.997010i	$0.475379\pi$
$402$	0	0
$403$	19.0226	0.947582
$404$	−5.86261	−0.291676
$405$	0	0
$406$	−2.21911	−0.110133
$407$	−11.0222	−0.546352
$408$	0	0
$409$	8.93617	0.441865	0.220933	−	0.975289i	$-0.429090\pi$
$409$	8.93617	0.441865	0.220933	+	0.975289i	$0.429090\pi$
$410$	0	0
$411$	0	0
$412$	−4.51450	−0.222413
$413$	−10.8710	−0.534927
$414$	0	0
$415$	0	0
$416$	17.4697	0.856520
$417$	0	0
$418$	10.9278	0.534495
$419$	−24.1237	−1.17852	−0.589260	−	0.807944i	$-0.700581\pi$
$419$	−24.1237	−1.17852	−0.589260	+	0.807944i	$0.700581\pi$
$420$	0	0
$421$	23.8602	1.16288	0.581438	−	0.813591i	$-0.302490\pi$
$421$	23.8602	1.16288	0.581438	+	0.813591i	$0.302490\pi$
$422$	13.3276	0.648779
$423$	0	0
$424$	−26.6721	−1.29531
$425$	0	0
$426$	0	0
$427$	14.4634	0.699933
$428$	−0.861848	−0.0416590
$429$	0	0
$430$	0	0
$431$	−4.45096	−0.214395	−0.107198	−	0.994238i	$-0.534188\pi$
$431$	−4.45096	−0.214395	−0.107198	+	0.994238i	$0.534188\pi$
$432$	0	0
$433$	−8.10929	−0.389707	−0.194854	−	0.980832i	$-0.562423\pi$
$433$	−8.10929	−0.389707	−0.194854	+	0.980832i	$0.562423\pi$
$434$	17.3494	0.832799
$435$	0	0
$436$	27.2423	1.30467
$437$	4.08401	0.195365
$438$	0	0
$439$	4.47180	0.213428	0.106714	−	0.994290i	$-0.465967\pi$
$439$	4.47180	0.213428	0.106714	+	0.994290i	$0.465967\pi$
$440$	0	0
$441$	0	0
$442$	−82.2643	−3.91291
$443$	9.60047	0.456132	0.228066	−	0.973646i	$-0.426760\pi$
$443$	9.60047	0.456132	0.228066	+	0.973646i	$0.426760\pi$
$444$	0	0
$445$	0	0
$446$	16.8142	0.796177
$447$	0	0
$448$	25.5071	1.20510
$449$	6.79944	0.320886	0.160443	−	0.987045i	$-0.448708\pi$
$449$	6.79944	0.320886	0.160443	+	0.987045i	$0.448708\pi$
$450$	0	0
$451$	7.05047	0.331994
$452$	−8.36795	−0.393595
$453$	0	0
$454$	59.7522	2.80431
$455$	0	0
$456$	0	0
$457$	19.5582	0.914894	0.457447	−	0.889237i	$-0.348764\pi$
$457$	19.5582	0.914894	0.457447	+	0.889237i	$0.348764\pi$
$458$	−7.62431	−0.356261
$459$	0	0
$460$	0	0
$461$	42.7081	1.98912	0.994558	−	0.104184i	$-0.0332230\pi$
$461$	42.7081	1.98912	0.994558	+	0.104184i	$0.0332230\pi$
$462$	0	0
$463$	7.42209	0.344934	0.172467	−	0.985015i	$-0.444826\pi$
$463$	7.42209	0.344934	0.172467	+	0.985015i	$0.444826\pi$
$464$	0.855693	0.0397245
$465$	0	0
$466$	44.3581	2.05485
$467$	−22.5041	−1.04136	−0.520682	−	0.853751i	$-0.674322\pi$
$467$	−22.5041	−1.04136	−0.520682	+	0.853751i	$0.674322\pi$
$468$	0	0
$469$	0.648506	0.0299452
$470$	0	0
$471$	0	0
$472$	18.8472	0.867510
$473$	8.71882	0.400892
$474$	0	0
$475$	0	0
$476$	−48.5347	−2.22458
$477$	0	0
$478$	−7.57241	−0.346354
$479$	10.1667	0.464530	0.232265	−	0.972653i	$-0.425386\pi$
$479$	10.1667	0.464530	0.232265	+	0.972653i	$0.425386\pi$
$480$	0	0
$481$	−58.4255	−2.66398
$482$	11.4197	0.520153
$483$	0	0
$484$	−35.6706	−1.62139
$485$	0	0
$486$	0	0
$487$	−34.9917	−1.58562	−0.792812	−	0.609467i	$-0.791384\pi$
$487$	−34.9917	−1.58562	−0.792812	+	0.609467i	$0.791384\pi$
$488$	−25.0753	−1.13511
$489$	0	0
$490$	0	0
$491$	2.27087	0.102483	0.0512415	−	0.998686i	$-0.483682\pi$
$491$	2.27087	0.102483	0.0512415	+	0.998686i	$0.483682\pi$
$492$	0	0
$493$	2.36795	0.106647
$494$	57.9249	2.60616
$495$	0	0
$496$	−6.68996	−0.300388
$497$	−31.8647	−1.42933
$498$	0	0
$499$	20.9929	0.939773	0.469887	−	0.882727i	$-0.344295\pi$
$499$	20.9929	0.939773	0.469887	+	0.882727i	$0.344295\pi$
$500$	0	0
$501$	0	0
$502$	5.34682	0.238640
$503$	−18.5041	−0.825055	−0.412528	−	0.910945i	$-0.635354\pi$
$503$	−18.5041	−0.825055	−0.412528	+	0.910945i	$0.635354\pi$
$504$	0	0
$505$	0	0
$506$	2.67575	0.118951
$507$	0	0
$508$	37.4263	1.66052
$509$	41.8472	1.85484	0.927421	−	0.374019i	$-0.122020\pi$
$509$	41.8472	1.85484	0.927421	+	0.374019i	$0.122020\pi$
$510$	0	0
$511$	21.9637	0.971619
$512$	−22.7423	−1.00508
$513$	0	0
$514$	26.2140	1.15625
$515$	0	0
$516$	0	0
$517$	7.19588	0.316475
$518$	−53.2867	−2.34128
$519$	0	0
$520$	0	0
$521$	−34.1103	−1.49440	−0.747199	−	0.664600i	$-0.768602\pi$
$521$	−34.1103	−1.49440	−0.747199	+	0.664600i	$0.768602\pi$
$522$	0	0
$523$	−36.4755	−1.59496	−0.797482	−	0.603343i	$-0.793835\pi$
$523$	−36.4755	−1.59496	−0.797482	+	0.603343i	$0.793835\pi$
$524$	59.4387	2.59659
$525$	0	0
$526$	−26.3180	−1.14752
$527$	−18.5131	−0.806442
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	34.1748	1.48167
$533$	37.3724	1.61878
$534$	0	0
$535$	0	0
$536$	−1.12432	−0.0485633
$537$	0	0
$538$	23.5701	1.01618
$539$	2.00535	0.0863765
$540$	0	0
$541$	28.2171	1.21315	0.606573	−	0.795028i	$-0.292544\pi$
$541$	28.2171	1.21315	0.606573	+	0.795028i	$0.292544\pi$
$542$	11.4866	0.493392
$543$	0	0
$544$	−17.0018	−0.728945
$545$	0	0
$546$	0	0
$547$	−15.1638	−0.648356	−0.324178	−	0.945996i	$-0.605088\pi$
$547$	−15.1638	−0.648356	−0.324178	+	0.945996i	$0.605088\pi$
$548$	−17.9423	−0.766456
$549$	0	0
$550$	0	0
$551$	−1.66735	−0.0710314
$552$	0	0
$553$	−10.2409	−0.435488
$554$	−17.9515	−0.762686
$555$	0	0
$556$	16.2633	0.689717
$557$	−9.52222	−0.403469	−0.201735	−	0.979440i	$-0.564658\pi$
$557$	−9.52222	−0.403469	−0.201735	+	0.979440i	$0.564658\pi$
$558$	0	0
$559$	46.2159	1.95472
$560$	0	0
$561$	0	0
$562$	−14.3065	−0.603484
$563$	32.7494	1.38022	0.690112	−	0.723702i	$-0.257561\pi$
$563$	32.7494	1.38022	0.690112	+	0.723702i	$0.257561\pi$
$564$	0	0
$565$	0	0
$566$	37.2185	1.56441
$567$	0	0
$568$	55.2441	2.31799
$569$	26.8926	1.12740	0.563698	−	0.825981i	$-0.309378\pi$
$569$	26.8926	1.12740	0.563698	+	0.825981i	$0.309378\pi$
$570$	0	0
$571$	0.920000	0.0385008	0.0192504	−	0.999815i	$-0.493872\pi$
$571$	0.920000	0.0385008	0.0192504	+	0.999815i	$0.493872\pi$
$572$	24.5498	1.02648
$573$	0	0
$574$	34.0853	1.42269
$575$	0	0
$576$	0	0
$577$	−42.5888	−1.77300	−0.886498	−	0.462732i	$-0.846869\pi$
$577$	−42.5888	−1.77300	−0.886498	+	0.462732i	$0.846869\pi$
$578$	39.6030	1.64727
$579$	0	0
$580$	0	0
$581$	24.7175	1.02545
$582$	0	0
$583$	7.57332	0.313655
$584$	−38.0787	−1.57571
$585$	0	0
$586$	−7.65462	−0.316209
$587$	9.51212	0.392607	0.196304	−	0.980543i	$-0.437106\pi$
$587$	9.51212	0.392607	0.196304	+	0.980543i	$0.437106\pi$
$588$	0	0
$589$	13.0356	0.537124
$590$	0	0
$591$	0	0
$592$	20.5474	0.844494
$593$	−31.0719	−1.27597	−0.637986	−	0.770048i	$-0.720232\pi$
$593$	−31.0719	−1.27597	−0.637986	+	0.770048i	$0.720232\pi$
$594$	0	0
$595$	0	0
$596$	10.1986	0.417753
$597$	0	0
$598$	14.1833	0.580000
$599$	−2.04065	−0.0833787	−0.0416893	−	0.999131i	$-0.513274\pi$
$599$	−2.04065	−0.0833787	−0.0416893	+	0.999131i	$0.513274\pi$
$600$	0	0
$601$	−25.2377	−1.02947	−0.514733	−	0.857351i	$-0.672109\pi$
$601$	−25.2377	−1.02947	−0.514733	+	0.857351i	$0.672109\pi$
$602$	42.1509	1.71794
$603$	0	0
$604$	43.7012	1.77818
$605$	0	0
$606$	0	0
$607$	−10.4305	−0.423361	−0.211680	−	0.977339i	$-0.567894\pi$
$607$	−10.4305	−0.423361	−0.211680	+	0.977339i	$0.567894\pi$
$608$	11.9715	0.485507
$609$	0	0
$610$	0	0
$611$	38.1432	1.54311
$612$	0	0
$613$	4.97745	0.201037	0.100519	−	0.994935i	$-0.467950\pi$
$613$	4.97745	0.201037	0.100519	+	0.994935i	$0.467950\pi$
$614$	82.0348	3.31066
$615$	0	0
$616$	10.1680	0.409681
$617$	30.5881	1.23143	0.615715	−	0.787969i	$-0.288867\pi$
$617$	30.5881	1.23143	0.615715	+	0.787969i	$0.288867\pi$
$618$	0	0
$619$	−48.0404	−1.93091	−0.965453	−	0.260579i	$-0.916087\pi$
$619$	−48.0404	−1.93091	−0.965453	+	0.260579i	$0.916087\pi$
$620$	0	0
$621$	0	0
$622$	44.6092	1.78866
$623$	12.9823	0.520123
$624$	0	0
$625$	0	0
$626$	−23.5908	−0.942878
$627$	0	0
$628$	−84.1030	−3.35608
$629$	56.8607	2.26718
$630$	0	0
$631$	10.6157	0.422605	0.211303	−	0.977421i	$-0.432229\pi$
$631$	10.6157	0.422605	0.211303	+	0.977421i	$0.432229\pi$
$632$	17.7547	0.706246
$633$	0	0
$634$	−21.9907	−0.873360
$635$	0	0
$636$	0	0
$637$	10.6298	0.421166
$638$	−1.09241	−0.0432488
$639$	0	0
$640$	0	0
$641$	−3.47844	−0.137390	−0.0686951	−	0.997638i	$-0.521884\pi$
$641$	−3.47844	−0.137390	−0.0686951	+	0.997638i	$0.521884\pi$
$642$	0	0
$643$	−2.05348	−0.0809813	−0.0404906	−	0.999180i	$-0.512892\pi$
$643$	−2.05348	−0.0809813	−0.0404906	+	0.999180i	$0.512892\pi$
$644$	8.36795	0.329743
$645$	0	0
$646$	−56.3734	−2.21798
$647$	−19.3408	−0.760367	−0.380184	−	0.924911i	$-0.624139\pi$
$647$	−19.3408	−0.760367	−0.380184	+	0.924911i	$0.624139\pi$
$648$	0	0
$649$	−5.35149	−0.210064
$650$	0	0
$651$	0	0
$652$	37.4015	1.46476
$653$	21.9288	0.858139	0.429070	−	0.903271i	$-0.358841\pi$
$653$	21.9288	0.858139	0.429070	+	0.903271i	$0.358841\pi$
$654$	0	0
$655$	0	0
$656$	−13.1433	−0.513161
$657$	0	0
$658$	34.7883	1.35619
$659$	−38.1351	−1.48553	−0.742767	−	0.669550i	$-0.766487\pi$
$659$	−38.1351	−1.48553	−0.742767	+	0.669550i	$0.766487\pi$
$660$	0	0
$661$	−28.9007	−1.12411	−0.562053	−	0.827101i	$-0.689988\pi$
$661$	−28.9007	−1.12411	−0.562053	+	0.827101i	$0.689988\pi$
$662$	−57.0419	−2.21700
$663$	0	0
$664$	−42.8529	−1.66302
$665$	0	0
$666$	0	0
$667$	−0.408263	−0.0158080
$668$	25.0867	0.970635
$669$	0	0
$670$	0	0
$671$	7.11993	0.274862
$672$	0	0
$673$	−49.5051	−1.90828	−0.954140	−	0.299361i	$-0.903226\pi$
$673$	−49.5051	−1.90828	−0.954140	+	0.299361i	$0.903226\pi$
$674$	−23.2390	−0.895131
$675$	0	0
$676$	82.5015	3.17313
$677$	6.87426	0.264199	0.132100	−	0.991236i	$-0.457828\pi$
$677$	6.87426	0.264199	0.132100	+	0.991236i	$0.457828\pi$
$678$	0	0
$679$	25.2776	0.970067
$680$	0	0
$681$	0	0
$682$	8.54064	0.327038
$683$	−36.9887	−1.41533	−0.707666	−	0.706547i	$-0.750252\pi$
$683$	−36.9887	−1.41533	−0.707666	+	0.706547i	$0.750252\pi$
$684$	0	0
$685$	0	0
$686$	47.7433	1.82285
$687$	0	0
$688$	−16.2535	−0.619657
$689$	40.1439	1.52936
$690$	0	0
$691$	38.5485	1.46645	0.733227	−	0.679984i	$-0.238013\pi$
$691$	38.5485	1.46645	0.733227	+	0.679984i	$0.238013\pi$
$692$	15.7375	0.598251
$693$	0	0
$694$	3.29472	0.125066
$695$	0	0
$696$	0	0
$697$	−36.3715	−1.37767
$698$	−76.8952	−2.91053
$699$	0	0
$700$	0	0
$701$	36.3146	1.37158	0.685791	−	0.727798i	$-0.259456\pi$
$701$	36.3146	1.37158	0.685791	+	0.727798i	$0.259456\pi$
$702$	0	0
$703$	−40.0374	−1.51004
$704$	12.5564	0.473239
$705$	0	0
$706$	27.2224	1.02453
$707$	3.65462	0.137446
$708$	0	0
$709$	−3.24463	−0.121855	−0.0609274	−	0.998142i	$-0.519406\pi$
$709$	−3.24463	−0.121855	−0.0609274	+	0.998142i	$0.519406\pi$
$710$	0	0
$711$	0	0
$712$	−22.5074	−0.843502
$713$	3.19187	0.119537
$714$	0	0
$715$	0	0
$716$	54.4539	2.03504
$717$	0	0
$718$	−23.6933	−0.884226
$719$	−10.4214	−0.388654	−0.194327	−	0.980937i	$-0.562252\pi$
$719$	−10.4214	−0.388654	−0.194327	+	0.980937i	$0.562252\pi$
$720$	0	0
$721$	2.81424	0.104808
$722$	−5.52338	−0.205559
$723$	0	0
$724$	48.1100	1.78799
$725$	0	0
$726$	0	0
$727$	9.67651	0.358882	0.179441	−	0.983769i	$-0.442571\pi$
$727$	9.67651	0.358882	0.179441	+	0.983769i	$0.442571\pi$
$728$	53.8976	1.99758
$729$	0	0
$730$	0	0
$731$	−44.9780	−1.66357
$732$	0	0
$733$	−15.5782	−0.575396	−0.287698	−	0.957721i	$-0.592890\pi$
$733$	−15.5782	−0.575396	−0.287698	+	0.957721i	$0.592890\pi$
$734$	−65.8715	−2.43136
$735$	0	0
$736$	2.93130	0.108049
$737$	0.319242	0.0117594
$738$	0	0
$739$	14.4328	0.530919	0.265459	−	0.964122i	$-0.414476\pi$
$739$	14.4328	0.530919	0.265459	+	0.964122i	$0.414476\pi$
$740$	0	0
$741$	0	0
$742$	36.6130	1.34411
$743$	−4.50305	−0.165201	−0.0826005	−	0.996583i	$-0.526323\pi$
$743$	−4.50305	−0.165201	−0.0826005	+	0.996583i	$0.526323\pi$
$744$	0	0
$745$	0	0
$746$	−68.3872	−2.50383
$747$	0	0
$748$	−23.8923	−0.873588
$749$	0.537257	0.0196309
$750$	0	0
$751$	18.5451	0.676721	0.338361	−	0.941017i	$-0.390128\pi$
$751$	18.5451	0.676721	0.338361	+	0.941017i	$0.390128\pi$
$752$	−13.4144	−0.489174
$753$	0	0
$754$	−5.79053	−0.210878
$755$	0	0
$756$	0	0
$757$	27.0366	0.982663	0.491332	−	0.870973i	$-0.336510\pi$
$757$	27.0366	0.982663	0.491332	+	0.870973i	$0.336510\pi$
$758$	−43.4840	−1.57941
$759$	0	0
$760$	0	0
$761$	−15.4066	−0.558490	−0.279245	−	0.960220i	$-0.590084\pi$
$761$	−15.4066	−0.558490	−0.279245	+	0.960220i	$0.590084\pi$
$762$	0	0
$763$	−16.9823	−0.614799
$764$	−51.2272	−1.85334
$765$	0	0
$766$	−69.1013	−2.49673
$767$	−28.3667	−1.02426
$768$	0	0
$769$	15.1216	0.545299	0.272650	−	0.962113i	$-0.412100\pi$
$769$	15.1216	0.545299	0.272650	+	0.962113i	$0.412100\pi$
$770$	0	0
$771$	0	0
$772$	−31.9217	−1.14889
$773$	−28.7131	−1.03274	−0.516370	−	0.856366i	$-0.672717\pi$
$773$	−28.7131	−1.03274	−0.516370	+	0.856366i	$0.672717\pi$
$774$	0	0
$775$	0	0
$776$	−43.8241	−1.57319
$777$	0	0
$778$	−6.81653	−0.244384
$779$	25.6103	0.917584
$780$	0	0
$781$	−15.6861	−0.561293
$782$	−13.8035	−0.493611
$783$	0	0
$784$	−3.73833	−0.133512
$785$	0	0
$786$	0	0
$787$	16.3131	0.581501	0.290750	−	0.956799i	$-0.406095\pi$
$787$	16.3131	0.581501	0.290750	+	0.956799i	$0.406095\pi$
$788$	−82.3905	−2.93504
$789$	0	0
$790$	0	0
$791$	5.21639	0.185473
$792$	0	0
$793$	37.7406	1.34021
$794$	−39.4088	−1.39857
$795$	0	0
$796$	33.0139	1.17015
$797$	2.88374	0.102147	0.0510736	−	0.998695i	$-0.483736\pi$
$797$	2.88374	0.102147	0.0510736	+	0.998695i	$0.483736\pi$
$798$	0	0
$799$	−37.1216	−1.31327
$800$	0	0
$801$	0	0
$802$	7.36523	0.260075
$803$	10.8121	0.381552
$804$	0	0
$805$	0	0
$806$	45.2714	1.59462
$807$	0	0
$808$	−6.33604	−0.222901
$809$	−39.7878	−1.39886	−0.699432	−	0.714699i	$-0.746564\pi$
$809$	−39.7878	−1.39886	−0.699432	+	0.714699i	$0.746564\pi$
$810$	0	0
$811$	38.5454	1.35351	0.676756	−	0.736208i	$-0.263385\pi$
$811$	38.5454	1.35351	0.676756	+	0.736208i	$0.263385\pi$
$812$	−3.41632	−0.119889
$813$	0	0
$814$	−26.2316	−0.919416
$815$	0	0
$816$	0	0
$817$	31.6705	1.10801
$818$	21.2670	0.743583
$819$	0	0
$820$	0	0
$821$	16.3428	0.570368	0.285184	−	0.958473i	$-0.407945\pi$
$821$	16.3428	0.570368	0.285184	+	0.958473i	$0.407945\pi$
$822$	0	0
$823$	−38.6446	−1.34707	−0.673534	−	0.739157i	$-0.735224\pi$
$823$	−38.6446	−1.34707	−0.673534	+	0.739157i	$0.735224\pi$
$824$	−4.87907	−0.169970
$825$	0	0
$826$	−25.8717	−0.900191
$827$	21.6969	0.754474	0.377237	−	0.926117i	$-0.376874\pi$
$827$	21.6969	0.754474	0.377237	+	0.926117i	$0.376874\pi$
$828$	0	0
$829$	13.9429	0.484259	0.242129	−	0.970244i	$-0.422154\pi$
$829$	13.9429	0.484259	0.242129	+	0.970244i	$0.422154\pi$
$830$	0	0
$831$	0	0
$832$	66.5579	2.30748
$833$	−10.3451	−0.358435
$834$	0	0
$835$	0	0
$836$	16.8233	0.581846
$837$	0	0
$838$	−57.4115	−1.98325
$839$	−52.4484	−1.81072	−0.905360	−	0.424644i	$-0.860399\pi$
$839$	−52.4484	−1.81072	−0.905360	+	0.424644i	$0.860399\pi$
$840$	0	0
$841$	−28.8333	−0.994252
$842$	56.7844	1.95692
$843$	0	0
$844$	20.5179	0.706255
$845$	0	0
$846$	0	0
$847$	22.2362	0.764046
$848$	−14.1180	−0.484815
$849$	0	0
$850$	0	0
$851$	−9.80345	−0.336058
$852$	0	0
$853$	−30.4634	−1.04305	−0.521524	−	0.853237i	$-0.674636\pi$
$853$	−30.4634	−1.04305	−0.521524	+	0.853237i	$0.674636\pi$
$854$	34.4211	1.17787
$855$	0	0
$856$	−0.931446	−0.0318362
$857$	−42.4911	−1.45147	−0.725735	−	0.687975i	$-0.758500\pi$
$857$	−42.4911	−1.45147	−0.725735	+	0.687975i	$0.758500\pi$
$858$	0	0
$859$	−15.5856	−0.531775	−0.265888	−	0.964004i	$-0.585665\pi$
$859$	−15.5856	−0.531775	−0.265888	+	0.964004i	$0.585665\pi$
$860$	0	0
$861$	0	0
$862$	−10.5927	−0.360790
$863$	50.8852	1.73215	0.866076	−	0.499913i	$-0.166635\pi$
$863$	50.8852	1.73215	0.866076	+	0.499913i	$0.166635\pi$
$864$	0	0
$865$	0	0
$866$	−19.2991	−0.655811
$867$	0	0
$868$	26.7094	0.906577
$869$	−5.04131	−0.171015
$870$	0	0
$871$	1.69220	0.0573382
$872$	29.4423	0.997041
$873$	0	0
$874$	9.71944	0.328765
$875$	0	0
$876$	0	0
$877$	30.1080	1.01667	0.508337	−	0.861158i	$-0.330260\pi$
$877$	30.1080	1.01667	0.508337	+	0.861158i	$0.330260\pi$
$878$	10.6424	0.359162
$879$	0	0
$880$	0	0
$881$	−20.2245	−0.681379	−0.340690	−	0.940176i	$-0.610661\pi$
$881$	−20.2245	−0.681379	−0.340690	+	0.940176i	$0.610661\pi$
$882$	0	0
$883$	30.6482	1.03139	0.515697	−	0.856771i	$-0.327533\pi$
$883$	30.6482	1.03139	0.515697	+	0.856771i	$0.327533\pi$
$884$	−126.646	−4.25956
$885$	0	0
$886$	22.8480	0.767592
$887$	14.0064	0.470290	0.235145	−	0.971960i	$-0.424443\pi$
$887$	14.0064	0.470290	0.235145	+	0.971960i	$0.424443\pi$
$888$	0	0
$889$	−23.3307	−0.782487
$890$	0	0
$891$	0	0
$892$	25.8855	0.866711
$893$	26.1385	0.874691
$894$	0	0
$895$	0	0
$896$	47.3139	1.58065
$897$	0	0
$898$	16.1818	0.539995
$899$	−1.30312	−0.0434616
$900$	0	0
$901$	−39.0687	−1.30157
$902$	16.7793	0.558688
$903$	0	0
$904$	−9.04370	−0.300789
$905$	0	0
$906$	0	0
$907$	36.4739	1.21109	0.605547	−	0.795809i	$-0.292954\pi$
$907$	36.4739	1.21109	0.605547	+	0.795809i	$0.292954\pi$
$908$	91.9884	3.05274
$909$	0	0
$910$	0	0
$911$	14.2327	0.471549	0.235775	−	0.971808i	$-0.424237\pi$
$911$	14.2327	0.471549	0.235775	+	0.971808i	$0.424237\pi$
$912$	0	0
$913$	12.1677	0.402693
$914$	46.5461	1.53961
$915$	0	0
$916$	−11.7376	−0.387822
$917$	−37.0528	−1.22359
$918$	0	0
$919$	34.3246	1.13226	0.566132	−	0.824315i	$-0.308439\pi$
$919$	34.3246	1.13226	0.566132	+	0.824315i	$0.308439\pi$
$920$	0	0
$921$	0	0
$922$	101.640	3.34734
$923$	−83.1474	−2.73683
$924$	0	0
$925$	0	0
$926$	17.6637	0.580464
$927$	0	0
$928$	−1.19674	−0.0392850
$929$	−6.14579	−0.201637	−0.100818	−	0.994905i	$-0.532146\pi$
$929$	−6.14579	−0.201637	−0.100818	+	0.994905i	$0.532146\pi$
$930$	0	0
$931$	7.28428	0.238733
$932$	68.2893	2.23689
$933$	0	0
$934$	−53.5569	−1.75244
$935$	0	0
$936$	0	0
$937$	−18.4581	−0.602998	−0.301499	−	0.953466i	$-0.597487\pi$
$937$	−18.4581	−0.602998	−0.301499	+	0.953466i	$0.597487\pi$
$938$	1.54337	0.0503927
$939$	0	0
$940$	0	0
$941$	−30.7623	−1.00282	−0.501411	−	0.865209i	$-0.667185\pi$
$941$	−30.7623	−1.00282	−0.501411	+	0.865209i	$0.667185\pi$
$942$	0	0
$943$	6.27087	0.204208
$944$	9.97615	0.324696
$945$	0	0
$946$	20.7497	0.674632
$947$	11.4692	0.372698	0.186349	−	0.982484i	$-0.440334\pi$
$947$	11.4692	0.372698	0.186349	+	0.982484i	$0.440334\pi$
$948$	0	0
$949$	57.3119	1.86042
$950$	0	0
$951$	0	0
$952$	−52.4541	−1.70005
$953$	2.40664	0.0779586	0.0389793	−	0.999240i	$-0.487589\pi$
$953$	2.40664	0.0779586	0.0389793	+	0.999240i	$0.487589\pi$
$954$	0	0
$955$	0	0
$956$	−11.6577	−0.377038
$957$	0	0
$958$	24.1956	0.781724
$959$	11.1848	0.361176
$960$	0	0
$961$	−20.8119	−0.671353
$962$	−139.046	−4.48301
$963$	0	0
$964$	17.5806	0.566234
$965$	0	0
$966$	0	0
$967$	−57.4766	−1.84832	−0.924162	−	0.382001i	$-0.875235\pi$
$967$	−57.4766	−1.84832	−0.924162	+	0.382001i	$0.875235\pi$
$968$	−38.5511	−1.23908
$969$	0	0
$970$	0	0
$971$	−53.1909	−1.70698	−0.853489	−	0.521111i	$-0.825518\pi$
$971$	−53.1909	−1.70698	−0.853489	+	0.521111i	$0.825518\pi$
$972$	0	0
$973$	−10.1382	−0.325015
$974$	−83.2759	−2.66833
$975$	0	0
$976$	−13.2728	−0.424853
$977$	54.4716	1.74270	0.871350	−	0.490661i	$-0.163245\pi$
$977$	54.4716	1.74270	0.871350	+	0.490661i	$0.163245\pi$
$978$	0	0
$979$	6.39080	0.204251
$980$	0	0
$981$	0	0
$982$	5.40440	0.172461
$983$	37.5908	1.19896	0.599480	−	0.800390i	$-0.295374\pi$
$983$	37.5908	1.19896	0.599480	+	0.800390i	$0.295374\pi$
$984$	0	0
$985$	0	0
$986$	5.63544	0.179469
$987$	0	0
$988$	89.1753	2.83704
$989$	7.75474	0.246587
$990$	0	0
$991$	14.0545	0.446455	0.223228	−	0.974766i	$-0.428341\pi$
$991$	14.0545	0.446455	0.223228	+	0.974766i	$0.428341\pi$
$992$	9.35635	0.297064
$993$	0	0
$994$	−75.8341	−2.40531
$995$	0	0
$996$	0	0
$997$	−10.9378	−0.346404	−0.173202	−	0.984886i	$-0.555411\pi$
$997$	−10.9378	−0.346404	−0.173202	+	0.984886i	$0.555411\pi$
$998$	49.9606	1.58148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.bv.1.4		4
3.2	odd	2		575.2.a.i.1.1		4
5.2	odd	4		1035.2.b.e.829.8		8
5.3	odd	4		1035.2.b.e.829.1		8
5.4	even	2		5175.2.a.bw.1.1		4
12.11	even	2		9200.2.a.cq.1.1		4
15.2	even	4		115.2.b.b.24.1	✓	8
15.8	even	4		115.2.b.b.24.8	yes	8
15.14	odd	2		575.2.a.j.1.4		4
60.23	odd	4		1840.2.e.d.369.2		8
60.47	odd	4		1840.2.e.d.369.7		8
60.59	even	2		9200.2.a.ck.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.1	✓	8	15.2	even	4
115.2.b.b.24.8	yes	8	15.8	even	4
575.2.a.i.1.1		4	3.2	odd	2
575.2.a.j.1.4		4	15.14	odd	2
1035.2.b.e.829.1		8	5.3	odd	4
1035.2.b.e.829.8		8	5.2	odd	4
1840.2.e.d.369.2		8	60.23	odd	4
1840.2.e.d.369.7		8	60.47	odd	4
5175.2.a.bv.1.4		4	1.1	even	1	trivial
5175.2.a.bw.1.1		4	5.4	even	2
9200.2.a.ck.1.4		4	60.59	even	2
9200.2.a.cq.1.1		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37988 q^{2} +3.66382 q^{4} -2.28394 q^{7} +3.95969 q^{8} +O(q^{10})\)	\(q+2.37988 q^{2} +3.66382 q^{4} -2.28394 q^{7} +3.95969 q^{8} -1.12432 q^{11} -5.95969 q^{13} -5.43550 q^{14} +2.09594 q^{16} +5.80007 q^{17} -4.08401 q^{19} -2.67575 q^{22} -1.00000 q^{23} -14.1833 q^{26} -8.36795 q^{28} +0.408263 q^{29} -3.19187 q^{31} -2.93130 q^{32} +13.8035 q^{34} +9.80345 q^{37} -9.71944 q^{38} -6.27087 q^{41} -7.75474 q^{43} -4.11931 q^{44} -2.37988 q^{46} -6.40020 q^{47} -1.78361 q^{49} -21.8352 q^{52} -6.73590 q^{53} -9.04370 q^{56} +0.971615 q^{58} +4.75976 q^{59} -6.33265 q^{61} -7.59627 q^{62} -11.1680 q^{64} -0.283942 q^{67} +21.2504 q^{68} +13.9516 q^{71} -9.61659 q^{73} +23.3310 q^{74} -14.9631 q^{76} +2.56788 q^{77} +4.48387 q^{79} -14.9239 q^{82} -10.8223 q^{83} -18.4553 q^{86} -4.45196 q^{88} -5.68414 q^{89} +13.6116 q^{91} -3.66382 q^{92} -15.2317 q^{94} -11.0676 q^{97} -4.24477 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8	\( 4 q + 2 q^{4} - 6 q^{7} + 6 q^{8} - 2 q^{11} - 14 q^{13} + 4 q^{14} + 2 q^{16} + 14 q^{17} - 4 q^{19} - 4 q^{22} - 4 q^{23} - 6 q^{26} - 18 q^{28} - 4 q^{29} + 2 q^{32} + 14 q^{34} - 2 q^{37} - 10 q^{38} + 8 q^{41} - 4 q^{43} - 6 q^{44} + 2 q^{47} - 18 q^{52} + 4 q^{53} - 14 q^{56} - 8 q^{61} - 28 q^{62} - 20 q^{64} + 2 q^{67} + 8 q^{68} + 24 q^{71} - 18 q^{73} + 36 q^{74} - 18 q^{76} + 4 q^{77} + 24 q^{79} - 32 q^{82} - 6 q^{83} - 14 q^{86} + 10 q^{88} + 8 q^{89} + 26 q^{91} - 2 q^{92} - 42 q^{94} - 34 q^{97} - 28 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 - 6 * q^7 + 6 * q^8 - 2 * q^11 - 14 * q^13 + 4 * q^14 + 2 * q^16 + 14 * q^17 - 4 * q^19 - 4 * q^22 - 4 * q^23 - 6 * q^26 - 18 * q^28 - 4 * q^29 + 2 * q^32 + 14 * q^34 - 2 * q^37 - 10 * q^38 + 8 * q^41 - 4 * q^43 - 6 * q^44 + 2 * q^47 - 18 * q^52 + 4 * q^53 - 14 * q^56 - 8 * q^61 - 28 * q^62 - 20 * q^64 + 2 * q^67 + 8 * q^68 + 24 * q^71 - 18 * q^73 + 36 * q^74 - 18 * q^76 + 4 * q^77 + 24 * q^79 - 32 * q^82 - 6 * q^83 - 14 * q^86 + 10 * q^88 + 8 * q^89 + 26 * q^91 - 2 * q^92 - 42 * q^94 - 34 * q^97 - 28 * q^98

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